Previous models of neuromodulation in cortical circuits have used either physiologically based networks of spiking neurons or simplified gain adjustments in low-dimensional connectionist models. Here we reduce a high-dimensional spiking neuronal network model, first to a four-population mean-field model and then to a two-population model. This provides a realistic implementation of neuromodulation in low-dimensional decision-making models, speeds up simulations by three orders of magnitude, and allows bifurcation and phase-plane analyses of the reduced models that illuminate neuromodulatory mechanisms. As modulation of excitation-inhibition varies, the network can move from unaroused states, through optimal performance to impulsive states, and eventually lose inhibition-driven winner-take-all behavior: all are clear outcomes of the bifurcation structure. We illustrate the value of reduced models by a study of the speed-accuracy tradeoff in decision making. The ability of such models to recreate neuromodulatory dynamics of the spiking network will accelerate the pace of future experiments linking behavioral data to cellular neurophysiology.